Confined Quantum Systems Using the Finite Element and Discrete Variable Representation Methods

Abstract

Confined quantum systems, particles or systems of particles that have their movements limited to a determined region of the space, have received attention since the beginning from the Quantum Theory. The physical and chemical properties of confined objects are modified with respect to free objects such as due the spatial confinement as to other factors as, for example, the electromagnetic field, not saturated chemical bounds, etc. In recent years, the interest in this area has grown sufficiently due the great set of phenomena and physical processes which can be characterized or be understood as confined quantum systems and that have various technological applications. The confined quantum systems are important, for example, in the embedding of atoms and molecules inside cavities such as zeolite molecular sieves, fullerenes, or solvent environments; in bubbles formed around foreign objects in the liquid helium or neutral plasma; in semiconductors structures in the mesoscopic-scale, as artificial atoms and molecules, or quantum dots; in atoms under pressure that are important for the agreement of the interior of planets, among others. Thus, different theoretical and computational methodologies have been used to study confined quantum systems. In particular, methods based on the variational formalism that expand the wave function in a set of basis functions as, for example, the discrete variable representation and the finite element method have been used successfully to treat systems with few electrons. In the present work our major aim is to present a review of applications of this class of methods to study typical confined quantum systems as the hydrogen atom and the two electron quantum dot, discussing the advantages and disadvantages of each one of them.